A269922 - OEIS

%I #32 Aug 10 2018 17:36:22

%S 21,483,483,6468,15018,6468,66066,258972,258972,66066,570570,3288327,

%T 5554188,3288327,570570,4390386,34374186,85421118,85421118,34374186,

%U 4390386,31039008,313530000,1059255456,1558792200,1059255456,313530000,31039008

%N Triangle read by rows: T(n,f) is the number of rooted maps with n edges and f faces on an orientable surface of genus 2.

%C Row n contains n-3 terms.

%H Gheorghe Coserea, <a href="/A269922/b269922.txt">Rows n = 4..204, flattened</a>

%H Sean R. Carrell, Guillaume Chapuy, <a href="http://arxiv.org/abs/1402.6300">Simple recurrence formulas to count maps on orientable surfaces</a>, arXiv:1402.6300 [math.CO], 2014.

%e Triangle starts:

%e n\f  [1]        [2]        [3]        [4]        [5]        [6]

%e [4]  21;

%e [5]  483,       483;

%e [6]  6468,      15018,     6468;

%e [7]  66066,     258972,    258972,    66066;

%e [8]  570570,    3288327,   5554188,   3288327,   570570;

%e [9]  4390386,   34374186,  85421118,  85421118,  34374186,  4390386;

%e [10] ...

%t Q[0, 1, 0] = 1; Q[n_, f_, g_] /; n<0 || f<0 || g<0 = 0;

%t Q[n_, f_, g_] := Q[n, f, g] = 6/(n+1)((2n-1)/3 Q[n-1, f, g] + (2n-1)/3 Q[n - 1, f-1, g] + (2n-3)(2n-2)(2n-1)/12 Q[n-2, f, g-1] + 1/2 Sum[l = n-k; Sum[v = f-u; Sum[j = g-i; Boole[l >= 1 && v >= 1 && j >= 0] (2k-1)(2l-1) Q[k-1, u, i] Q[l-1, v, j], {i, 0, g}], {u, 1, f}], {k, 1, n}]);

%t Table[Q[n, f, 2], {n, 4, 10}, {f, 1, n-3}] // Flatten (* _Jean-François Alcover_, Aug 10 2018 *)

%o (PARI)

%o N = 10; G = 2; gmax(n) = min(n\2, G);

%o Q = matrix(N + 1, N + 1);

%o Qget(n, g) = { if (g < 0 || g > n/2, 0, Q[n+1, g+1]) };

%o Qset(n, g, v) = { Q[n+1, g+1] = v };

%o Quadric({x=1}) = {

%o   Qset(0, 0, x);

%o   for (n = 1, length(Q)-1, for (g = 0, gmax(n),

%o     my(t1 = (1+x)*(2*n-1)/3 * Qget(n-1, g),

%o        t2 = (2*n-3)*(2*n-2)*(2*n-1)/12 * Qget(n-2, g-1),

%o        t3 = 1/2 * sum(k = 1, n-1, sum(i = 0, g,

%o        (2*k-1) * (2*(n-k)-1) * Qget(k-1, i) * Qget(n-k-1, g-i))));

%o     Qset(n, g, (t1 + t2 + t3) * 6/(n+1))));

%o };

%o Quadric('x);

%o concat(apply(p->Vecrev(p/'x), vector(N+1 - 2*G, n, Qget(n-1 + 2*G, G))))

%Y Columns f=1-10 give: A006298 f=1, A288082 f=2, A288083 f=3, A288084 f=4, A288085 f=5, A288086 f=6, A288087 f=7, A288088 f=8, A288089 f=9, A288090 f=10.

%Y Row sums give A006301 (column 2 of A269919).

%Y Cf. A006299 (row maxima), A269921.

%K nonn,tabl

%O 4,1

%A _Gheorghe Coserea_, Mar 15 2016